Beyond the Descartes Circle Theorem

The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b1 + b 2 2 + b 2 3 + b 2 4). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes Theorem. These relations have elegant matrix generalizations to the n-dimensional case, in each of Euclidean, spherical, and hyperbolic geometries. These include analogues of the Descartes circle theorem for spherical and hyperbolic space.